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arithmetic surfaces and their

regular models

Dissertation zur Erlangung des Doktorgrades Dr. rer. nat.

der Fakult¨at fur¨ Mathematik und Wirtschaftswissenschaften

der Universit¨at Ulm

Vorgelegt von Franz Johannes Kir´aly aus Ulm

Ulm, 2010Dekan: Prof. Dr. Werner Kratz

Gutachter: Prof. Dr. Werner Lu¨tkebohmert

Datum der Promotion: 02. November 2010Contents

1 Introduction 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 About notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Tame quotient singularities 4

2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Basic facts on invariant rings . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Excellent rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.4 Tame local actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.5 Toric schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 Extension to mixed characteristic . . . . . . . . . . . . . . . . . . . . 17

2.7 Desingularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.8 A characterization of toric surface singularities . . . . . . . . . . . . 24

3 p-cyclic group actions on local normal rings 26

3.1 Motivating examples in dimension 2 . . . . . . . . . . . . . . . . . . 26

3.2 A criterion for monogeneity . . . . . . . . . . . . . . . . . . . . . . . 32

3.3 Application to models of curves . . . . . . . . . . . . . . . . . . . . . 38

4 Wild quotient singularities of surfaces 46

4.1 Brief overview on chapter 4 . . . . . . . . . . . . . . . . . . . . . . . 46

4.2 Models of local rings . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.1 Deﬁnition of models . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.2 Zariski valuations . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2.3 Introducing the G -action . . . . . . . . . . . . . . . . . . . . 56

4.2.4 The graph structure of models and the G -action . . . . . . . 60

4.2.5 Minimal regular realization of a valuation . . . . . . . . . . . 63

4.2.6 Rings of components and parameters . . . . . . . . . . . . . . 66

I4.2.7 Application to components with K -rational center . . . . . 67

4.3 Resolution of wild quotient singularities . . . . . . . . . . . . . . . . 70

4.3.1 Examining the naive approach . . . . . . . . . . . . . . . . . 70

4.3.2 Critical components and correspondence of models . . . . . . 73

4.3.3 The augmentation ideal on critical components . . . . . . . . 78

4.3.4 Application to components with K -rational center . . . . . 79

4.4 Tame descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.4.1 Galois extension and tame descent . . . . . . . . . . . . . . . 85

4.4.2 Tame invariance of critical components . . . . . . . . . . . . 89

4.4.3 Quotients of stable models. . . . . . . . . . . . . . . . . . . . 90

Bibliography 93

IIChapter 1

Introduction

1.1 Introduction

Resolutionofsingularitiesisoneoftheoldestandyetstilloneofthemostintriguing

topics in mathematics. Ever since Isaac Newton’s calculations on Puiseux series

expansionsofcurves,ithasbeenofmajorinteresttodescribeorremovesingularities

on varieties. The ﬁrst rigorous proof for desingularization of surfaces over the

complexnumberswasgiven1934byWalker[Wal34]. Afewyearslater, Zariskigave

proofsfordesingularizationofsurfaces[Zar39]andthree-folds[Zar44]overarbitrary

ﬁelds of characteristic zero. In 1964, Hironaka gave his famous proof for reduced

schemes of ﬁnite type over ﬁelds of characteristic zero [Hir64], Abhyankar presented

hisresultforresolutionoverﬁeldsofcharacteristicsevenorgreaterin1966[Abh66].

In 1978, Lipman proved desingularization for quasi-excellent surfaces which also

include arithmetic surfaces [Lip78]. For the remaining cases, desingularization is

still an open problem despite many eﬀorts; however, Grothendieck has proved that

quasi-excellency is a necessary condition for a locally Noetherian scheme to have

a desingularization [Gro65, 7.9]. By Lipman’s result, we know that it is also a

suﬃcient condition for dimension two and less.

This thesis is motivated by the study of quotient singularities, occurring on the

scheme of orbits by a ﬁnite group action on a regular scheme, the so-called quotient

scheme. Quotient singularities are an important class of singularities in any char-

acteristic which canonically arise in context of actions on schemes. The problem

of quotient singularities was ﬁrst studied by Jung [Jun08] and Hirzebruch [Hir53]

for a cyclic group acting on a regular complex surface. Here the quotient surface

exhibits the famous Hirzebruch-Jung singularities; their resolution combinatorics

can be related directly to the action of the group G. General groups acting on

complex varieties were later examined by Cartan [Car57] and Brieskorn [Bri68].

When one generalizes the problem to arbitrary (locally Noetherian quasi-excellent)

schemes, one has to note that the problem of quotient singularities is essentially of

local nature, so it is merely a question in commutative algebra. One can reduce to

the following situation:

Let B a Noetherian local regular ring, and G,→ AutB a ﬁnite group action on

B. Denote by k the residue ﬁeld of B, by p = chark its characteristic. The

Gring of invariants A =B is the subring of B consisting exactly of the elements

invariant under G and it can happen that A is not regular. Here B plays the

role of a local germ of the scheme in consideration, taken at a closed point; and A

1is the corresponding germ on the quotient scheme. The main goal in the theory of

quotient singularities is to relate the structure of A , e.g. the combinatorics of its

minimal desingularization, to the group action of G on B.

The case where p is coprime to #G, the so-called tame case, has been extensively

studied in literature. In this context, the Hirzebruch-Jung singularities over the

complex numbers can be generalized with minor eﬀorts to the case of a cyclic group

acting on varieties over arbitrary ﬁelds of characteristic zero. The same can be

done for arbitrary group actions. There have been also some ad-hoc adaptations of

theseresultstomoregeneral B mostlyindimension 2 , e.g. thecaseofarithmetic

surfacesin[Vie77]. Moreover,inthetamecase,acelebratedtheoremofSerre[Ser68]

classiﬁesexactlywhenthering A isregularintermsof G, namelyifandonlyif G

is generated by so-called pseudo-reﬂections. There seems to be consensus that the

tame case is relatively well understood, and that those results can be generalized to

arbitraryringsinthetamecase. However, asof2010, thereisnostandardreference

in literature which goes beyond particular applications.

The case where p divides #G is called the wild case, since the classical methods

from the tame case fail by elementary means. To the author’s knowledge, the only

results concern the simplest non-trivial case where G =Z/pZ, and B is regular

of very speciﬁc form: M. Artin [Art75] has obtained a few results for p = 2 and

B =k[[X ,X ]]. InPeskin’sthesis, [Pes83]severalbasicresultsaboutthewildcase1 2

are collected and the results of Artin are generalized for speciﬁc group actions with

p≥ 3.

There seems to have been little progress after that until Lorenzini’s unpublished

paper [Lor06], in which some structural results on quotient singularities with focus

on quotients of stable models of curves by prime cyclic actions are obtained. The

biggest part of those results uses the global machinery of N´eron models and many

ad-hoccombinatorialconstructionswhichunfortunatelygivesnoinsightonhowthe

G -action relates to the structure of the singularities. However, in the context of

the local results of Artin these results suggest that it might be possible to obtain

similar structural results only with local methods.

The goal of this thesis is to understand the relation between B and A in terms

of the group action G on B, in the simplest non-trivial case where G =hσi is

prime cyclic.

In chapter 2, we collect and extend classical results on tame quotient singularities

and discuss them in the context of toric geometry.

In chapter 3, we prove an algebraic result about the invariant morphism. We prove

that B is a monogenous A -algebra if and only if the augmentation ideal

I ={(σ−id)b ; b∈B}G

of B isprincipal. Ifinparticular B isregular, thisimpliesthat A isalsoregular.

In chapter 4, we assume that B is a local regular normal crossings germ of an

arithmeticsurfaceoverthespectrumofacompletediscretevaluationring R. Using

birationalgeometry,combinatorialmethodsandtheresultsfromchapter3,werelate

the structure of the minimal normal crossings desingularization of A to the group

action of G on B. Furthermore, we examine the behavior of the singularity of A

with respect to tame base extension.

A more detailed overview on the content and the utilized methods can be found at

the beginning of each chapter.

21.2 About notation

Inthisthesiswewilltrytousecommonsymbolsandnotations. However,wewantto

pointoutseveralthingswhichmightbesourcesofconfusionsincethecorresponding

notation is not uniform worldwide. At most occasions, this will be also said in the

text.

For a ring C, we will denote by Q(C) its ﬁeld of fractions. If C is local, then

bC will denote the completion of C by the topology given by its maximal ideal.

Forascheme S andaclosedpoint s on S, wewilldenoteby K(S) thefunction

ﬁeld of S , and by k(s) the residue ﬁeld of S at s. The structure sheaf of S

will be denoted by O .S

∼Isomorphies will be denoted by , congruences by ≡.=

The empty set will be denoted by ∅.

The symbol ∂ will always be used to denote boundaries. Partial derivatives do not

occur in this thesis.

Thegenerallineargroupofdimension n overaﬁeld k willbedenotedby GL (k).n

Similarly, its special subgroup will be denoted SL (k).n

The natural numbers will be denoted by N and contain zero. By Q we will≥0

denote the nonnegative rational numbers.

1.3 Acknowledgments

I would like to express my gratitude to my advisor Prof. Werner Lut¨ kebohmert,

whose expertise and understanding have supported me during the course of my

thesis. I especially want to thank him for all the time he has sacriﬁced for me,

especially in the critical phases, as well as for many of the helpful suggestions and

fruitful discussions.

Also, I want to thank Prof. Stefan Wewers for the discussions and suggestions con-

cerningmyworkwhichhavebroughtforththenecessarymomentumofcreativityto

handlemythesis,andProf.IreneBouwforthecontinuedwillingnesstosupportively

discuss the course of my work. I would also like to thank both for their continu-

ing and ongoing pursuit which has taught me to learn and develop mathematics in

increasing self-reliance.

I want to thank the colleagues at the institute in Ulm which have supported me

during all those years, especially Dr. Louis Brewis from and with whom I have

learnt much of my basic knowledge in Algebraic Geometry. I wish him all the best

in his new exciting ﬁeld of activity.

Furthermore,IwanttothanktheUlmGraduateSchoolonMathematicalAnalysisof

Evolution, Information and Complexity, the German National Merit Foundation,

and the TU Berlin Machine Learning Group for their ﬁnancial and non-material

support in those years.

Last but not least, I want to thank my family for the incessant and unconditional

support in the time of my thesis, and of course all the other people who have stood

on my side.

3Chapter 2

Tame quotient singularities

2.1 Overview

In this chapter we will summarize some classical facts on invariant rings. We will

consider a Noetherian local normal ring B with ﬁnite group G,→ AutB acting

Gon it, and try to understand the structure of the invariant ring A =B .

In section 2.2, Basic facts on invariant rings, we introduce the setting for this

chapter in detail and make some basic deﬁnitions. In section 2.3, Excellent rings,

we state some classical results about excellent rings, which will allow us under

certain conditions to reduce to the case where B and A are complete.

In section 2.4, Tame local actions, we will review classical results in the case where

the action of G on B is tame, i.e. when #G is coprime to the residue charac-

teristic of B. We begin with classical linearization results for general tame group

actions, utlizing the original argument of Cartan [Car57]. Then we concentrate on

cyclic actions and explicitly describe the structure of the ring of invariants A , cf.

Propositions 2.4.12 and 2.4.10. We will also relate those results to Serre’s theorem

on pseudo-reﬂections and regular rings, cf. Corollary 2.4.17.

The next objective will be to obtain the desingularization in the case where B

is regular. If B is of dimension 2 , this will lead us for example to the classi-

cal Hirzebruch-Jung singularities as in [Jun08] and [Hir53]. Instead of doing the

calculations explicitly, we will introduce the notion of toric schemes and rings to

avoid lengthy calculations obscuring the structural intuition, since it turns out that

tame cyclic quotient singularities are toric. In section 2.5, Toric schemes, we will

brieﬂy introduce toric geometry using the book of Fulton [Ful93] and formulate our

probleminthissetting. Asdonealreadyfrequentlybyseveralauthors, wewillwork

over an arbitrary base ﬁeld instead of C.

In section 2.6, Extension to mixed characteristic, we extend the classic toric geome-

try over ﬁelds to arbitrary characteristic in the vein of Mumford. However, we have

to broaden Mumford’s setting in [KKMSD70, Chapter IV, §3] a bit; this allows us

for example to also treat rings like R[[X ,X ]]/(X X −π); for this we will deﬁne1 2 1 2

the concept of a locally toric ring 2.6.2, which is the local equivalent of Kato’s con-

cept of log-regularity, cf. [Kat94, §3]. We refer the reader to the beginning of this

section for a more detailed discussion.

In section 2.7, Desingularization, we will use the toric theory to desingularize the

tame cyclic quotient singularities. We will do this by describing a method to desin-

4